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Theorem mexval 31998
 Description: The set of expressions, which are pairs whose first element is a typecode, and whose second element is a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mexval.k 𝐾 = (mTC‘𝑇)
mexval.e 𝐸 = (mEx‘𝑇)
mexval.r 𝑅 = (mREx‘𝑇)
Assertion
Ref Expression
mexval 𝐸 = (𝐾 × 𝑅)

Proof of Theorem mexval
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 mexval.e . 2 𝐸 = (mEx‘𝑇)
2 fveq2 6446 . . . . . 6 (𝑡 = 𝑇 → (mTC‘𝑡) = (mTC‘𝑇))
3 mexval.k . . . . . 6 𝐾 = (mTC‘𝑇)
42, 3syl6eqr 2832 . . . . 5 (𝑡 = 𝑇 → (mTC‘𝑡) = 𝐾)
5 fveq2 6446 . . . . . 6 (𝑡 = 𝑇 → (mREx‘𝑡) = (mREx‘𝑇))
6 mexval.r . . . . . 6 𝑅 = (mREx‘𝑇)
75, 6syl6eqr 2832 . . . . 5 (𝑡 = 𝑇 → (mREx‘𝑡) = 𝑅)
84, 7xpeq12d 5386 . . . 4 (𝑡 = 𝑇 → ((mTC‘𝑡) × (mREx‘𝑡)) = (𝐾 × 𝑅))
9 df-mex 31983 . . . 4 mEx = (𝑡 ∈ V ↦ ((mTC‘𝑡) × (mREx‘𝑡)))
10 fvex 6459 . . . . 5 (mTC‘𝑡) ∈ V
11 fvex 6459 . . . . 5 (mREx‘𝑡) ∈ V
1210, 11xpex 7240 . . . 4 ((mTC‘𝑡) × (mREx‘𝑡)) ∈ V
138, 9, 12fvmpt3i 6547 . . 3 (𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅))
14 xp0 5806 . . . . 5 (𝐾 × ∅) = ∅
1514eqcomi 2787 . . . 4 ∅ = (𝐾 × ∅)
16 fvprc 6439 . . . 4 𝑇 ∈ V → (mEx‘𝑇) = ∅)
17 fvprc 6439 . . . . . 6 𝑇 ∈ V → (mREx‘𝑇) = ∅)
186, 17syl5eq 2826 . . . . 5 𝑇 ∈ V → 𝑅 = ∅)
1918xpeq2d 5385 . . . 4 𝑇 ∈ V → (𝐾 × 𝑅) = (𝐾 × ∅))
2015, 16, 193eqtr4a 2840 . . 3 𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅))
2113, 20pm2.61i 177 . 2 (mEx‘𝑇) = (𝐾 × 𝑅)
221, 21eqtri 2802 1 𝐸 = (𝐾 × 𝑅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1601   ∈ wcel 2107  Vcvv 3398  ∅c0 4141   × cxp 5353  ‘cfv 6135  mTCcmtc 31960  mRExcmrex 31962  mExcmex 31963 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-mex 31983 This theorem is referenced by:  mexval2  31999  msubff  32026  msubco  32027  msubff1  32052  mvhf  32054  msubvrs  32056
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